Question

1. match the graph with an appropriate equation. choices are listed below the graphs. graph a graph b graph c graph d functions y = 2sin(x) y = sin(2x) y = sin(x) + 2 y = 2cos(x) y = cos(2x) y = cos(x) + 2

Explanation:

Step1: Analyze \( y = 2\cos(x) \)

The general form of a cosine function is \( y = A\cos(Bx - C) + D \). For \( y = 2\cos(x) \), \( A = 2 \) (amplitude is 2), \( B = 1 \) (period \( \frac{2\pi}{B}=2\pi \)), \( C = 0 \), \( D = 0 \). The amplitude of \( \cos(x) \) is 1, so doubling the amplitude (to 2) will stretch the graph vertically. The period remains \( 2\pi \). Looking at the graphs, Graph A has a vertical stretch (amplitude 2) and period \( 2\pi \), matching \( y = 2\cos(x) \).

Step2: Analyze \( y = \cos(2x) \)

For \( y = \cos(2x) \), \( A = 1 \), \( B = 2 \), so period \( \frac{2\pi}{2}=\pi \). The amplitude is 1 (same as \( \cos(x) \)), but the period is halved (compressed horizontally). Graph C has a period of \( \pi \) (more cycles in \( 2\pi \) interval) and amplitude 1, matching \( y = \cos(2x) \).

Step3: Analyze \( y = \cos(x) + 2 \)

For \( y = \cos(x) + 2 \), \( A = 1 \), \( B = 1 \) (period \( 2\pi \)), \( D = 2 \) (vertical shift up by 2). The graph should be the standard \( \cos(x) \) shifted up, with amplitude 1 and period \( 2\pi \). Graph D is shifted up (y - values around 2 - 4) with period \( 2\pi \) and amplitude 1, matching \( y = \cos(x) + 2 \).

Step4: Analyze \( y = 2\sin(x) \)

The general form of a sine function is \( y = A\sin(Bx - C) + D \). For \( y = 2\sin(x) \), \( A = 2 \) (amplitude 2), \( B = 1 \) (period \( 2\pi \)). The sine function starts at 0 (unlike cosine which starts at 1). Graph B has amplitude 2, period \( 2\pi \), and starts at 0 (sine - like behavior), matching \( y = 2\sin(x) \).

Step5: Analyze \( y = \sin(2x) \)

For \( y = \sin(2x) \), \( A = 1 \), \( B = 2 \), period \( \pi \). The sine function has amplitude 1, period \( \pi \) (compressed horizontally). The remaining graph (after matching others) with period \( \pi \), amplitude 1, and sine - like behavior (starts at 0) matches \( y = \sin(2x) \).

Step6: Analyze \( y = \sin(x) + 2 \)

For \( y = \sin(x) + 2 \), \( A = 1 \), \( B = 1 \) (period \( 2\pi \)), \( D = 2 \) (vertical shift up by 2). The graph should be \( \sin(x) \) shifted up, amplitude 1, period \( 2\pi \). If not already matched, but based on the above, we assume the remaining matches are as above.

Answer:

• \( y = 2\cos(x) \): Graph A
• \( y = \cos(2x) \): Graph C
• \( y = \cos(x) + 2 \): Graph D
• \( y = 2\sin(x) \): Graph B
• \( y = \sin(2x) \): (remaining graph with period \( \pi \), sine - like, amplitude 1)
• \( y = \sin(x) + 2 \): (if not matched, but based on typical matching, the above are the main matches for the cosine - related and \( 2\sin(x) \))

(Note: The exact matching depends on the full set of graphs, but the above is the analysis for each function's transformation from the parent cosine/sine functions.)